# RE: OFV higher with FOCEI than FO

From: Wang, Yaning <yaning.wang>
Date: Wed, 10 Dec 2008 20:44:30 -0500

Matt:
That's not true. Those two references are discussing when the linearized
structure model can also be derived from direct Laplacian approximation
of the marginal likelihood. When there is an interaction between
residual and between subject variability (or residual error model
contain subject-specific random effect), linearizing the structure model
around eta_hat cannot be derived from the Laplacian approximation any
more. But in NONMEM, FOCE with interaction (when residual error model
contain subject-specific random effect) is still derived from Laplacian
approximation. In other words, NONMEM does not linearize the structure
model for FOCE with interaction case. I discussed this in details in my
paper (1). Adding the following splus code to the splus code in my paper
and using the simple numerical example, you can see how NONMEM is
calculating the objective function for FOCE with interaction. These
things are further visualized in my talk recently put on ACCP webpage
(http://www.accp1.org/pharmacometrics/PopPKCourse.html).

Yaning

#reproduce NONMEM result using my equation 28 which is further
approximation of Laplacian method
sum<-0
for (i in 1:10) {
data1<-data[data\$ID==i,]
cov<-data1\$fp%*%t(data1\$fp)*omega+diag(data1\$f**2)*eps+2*data1\$fp%*%t(da
ta1\$fp)*omega*eps
cov1<-diag(data1\$f**2)*eps
ginv<-solve(cov1)
sec<-t(data1\$DV-data1\$IPRE)%*%ginv%*%(data1\$DV-data1\$IPRE)+data1\$ETA1[1]
**2/omega
frs<-determinant(cov, logarithm=T)\$modulus[[1]]
sum1<-sec+frs
sum<-sum+sum1
}
sum#39.45756 same as NONMEM OFV 39.458

1. Yaning Wang. Derivation of various NONMEM estimation methods. Journal
of Pharmacokinetics and pharmacodynamics. 34:575-93 (2007)

Yaning Wang, Ph.D.
Office of Clinical Pharmacology
Office of Translational Science
Center for Drug Evaluation and Research
Phone: 301-796-1624
Email: yaning.wang
"The contents of this message are mine personally and do not necessarily
reflect any position of the Government or the Food and Drug

________________________________

From: owner-nmusers
On Behalf Of Matt Hutmacher
Sent: Wednesday, December 10, 2008 2:04 PM
To: 'Bob Leary'; ayyappa.5.chaturvedula
owner-nmusers
Subject: RE: [NMusers] OFV higher with FOCEI than FO

Hi Bob,

I would just add one point of clarification. My understanding is that
the FOCE approximate is a Laplace-based approximation (related to it)
only if the within subject residual error model does not contain any
subject-specific random effects.

Wolfinger R (1993). Laplace's approximation for nonlinear mixed models.
Biometrika 80, 791-795.

Vonesh ER, Chinchilli VM (1997). Linear and nonlinear models for the
analysis of repeated measurements. Marcel Dekker.

Matt

From: owner-nmusers
On Behalf Of Bob Leary
Sent: Wednesday, December 10, 2008 12:11 PM
To: ayyappa.5.chaturvedula
nmusers
Subject: RE: [NMusers] OFV higher with FOCEI than FO

As shown by X. Wang, FO, FOCE and LAPLACE form a hierarchy of
approximations.

Both the FO and FOCE methods are based on the same underlying Laplacian
approximation to the

integral of the joint likelihood function of the random effects (eta's).

The basic Laplace approximation requires knowledge of

the value of the joint likelihood function at its peak, and the second
derivatives at the

eta values at which the peak is reached.

Hessian matrix of second derivatives at the peak of the joint likelihood
function

from first derivatives, but accurately

determines the position of the peak (the empirical Bayes estimates)

in random effects (eta) space

and the function value at the peak (this determination of the EBE's is
what the 'conditional step'

is all about and is computationally costly.)

Although the underlying Laplacian approximation is based on the local
behavior of the

joint log likelihood function in the neighborhood of its peak, FO does
not investigate the behavior

of the joint likelihood function near its peak at all (which is
basically why FO estimates can be arbitrarily

poor). Instead it guestimates the value at the peak by extrapolating
from eta=0, using a single Newton step

based on approximate first and second derivatives at eta=0. It also
simply assigns the FOCE

approximate values of the

second derivatives at eta=0 to the values at the peak in order to
evaluate the Laplacian approximation.

These additional approximations layered on top of the basic Laplacian
and FOCE approximations

by FO are quite dubious for significantly nonlinear model functions, and
often result in very poor quality

parameter estimates compared to FOCE and Laplace.

Strictly speaking. FOCE and FO objective values cannot be compared in
any consistently meaningful sense.

But loosely speaking, since both FO and FOCE share a common base
Laplacian approximation, but FO layers

on additional approximations on top of FOCE, the difference in FO vs
FOCE objective values reflects the

effects of the additional FO approximations. Large differences may
suggest that the additional FO approximations

have large effects, and make the FO estimates even more suspect relative
to FOCE.

Robert H. Leary, PhD
Principal Software Engineer
Pharsight Corp.
5520 Dillard Dr., Suite 210
Cary, NC 27511

Phone/Voice Mail: (919) 852-4625, Fax: (919) 859-6871

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-----Original Message-----
From: owner-nmusers
[mailto:owner-nmusers
ayyappa.5.chaturvedula
Sent: Wednesday, December 10, 2008 9:40 AM
To: owner-nmusers
Subject: [NMusers] OFV higher with FOCEI than FO

Dear All,

I am analyzing a data set pooled from 4 clinical studies with
rich sampling. When I fit a 2 comp oral absorption model with lag time
using FO, I got successful minimization with COV step, but minimization
was not successful when I used FO parameter estimates as initial
estimates for FOCE run. When I used FOCE with INTER minimization was
successful with COV step but the OFV is much higher (~25000 vs 20000)
with FOCEI estimation than FO. The parameter estimates make more sense
with FOCEI than FO. My questions are,

Can we get something like this or I am missing something here?

Can we compare OFV between different estimation methods (my
understanding is no and OFV in case of FO does not make a lot of sense)?

Regards,
Ayyappa Chaturvedula
GlaxoSmithKline